What exactly is "Psik Reisha"? I know this has something to do with the laws of Shabbat, and possibly something to do with unintended consequences that are allowed, but what exactly is it, and in what kinds of situations is it applicable?

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Psik Reishei refers to an inevitable consequence of an action. Literally "cutting off its head", the original example (see Rambam Shabbos 1:6) is someone who wants to give his child a chicken head to play with on Shabbos (don't ask, I guess they did those things back then). The problem is killing a chicken on Shabbos is forbidden.

Now, on Shabbos it is sometimes permitted to do an action that might have a forbidden consequence, if it is not one's intent to cause the forbidden consequence. This is not the case with psik reishei. If you cut off the chicken's head, the unintended consequence (the chicken's death) is inevitable, and so - forbidden.

Examples of psik reishei that I've heard include dragging a very heavy object through dirt (guaranteed to cause a groove in the ground = plowing). A lighter object would not be psik reishei since it's not certain that it will create a groove. (Magen Avraham to OC 337:1)

In a case where the inevitable consequence is an undesired one, it is called psik reishei d'lo nicha leih:

[O]n Shabbos, where the resulting consequence is neither wanted nor intended, it is classified as a psik reisha d’lo nicha leih, which is permitted by Torah law but prohibited by Rabbinic law. -Star-K.org